The use of strained substrates may overcome indium incorporation limits without inducing plastic relaxation in InGaN quantum wells, and this is particularly important for short-period InGaN/GaN superlattices. By incorporating elastic strain into these heterostructures, their optoelectronic behavior is modified. Our study employed density functional theory calculations to investigate the variation in the band-gap energy of short-period InGaN/GaN superlattices that comprise pseudomorphic quantum wells with a thickness of just one monolayer. Heterostructures with equibiaxially strained GaN barriers were compared with respective ones with relaxed barriers. The findings reveal a reduction of the band gap for lower indium contents, which is attributed to the influence of the highly strained nitrogen sublattice. However, above mid-range indium compositions, the situation is reversed, and the band gap increases with the indium content. This phenomenon is attributed to the reduction of the compressive strain in the quantum wells caused by the tensile strain of the barriers. Our study also considered local indium clustering induced by phase separation as another possible modifier of the band gap. However, unlike the substrate-controlled strain, this was not found to exert a significant influence on the band gap. Overall, this study provides important insights into the behavior of the band-gap energy of strained superlattices toward optimizing the performance of optoelectronic devices based on InGaN/GaN heterostructures.